Find one value of $x$ that is a solution to the equation: $(x^2+3)^2=4x^2+12$ $x=$
We could solve for $x$ by expanding $(x^2+3)^2$, combining terms that are alike, and using the quadratic formula or factoring to solve for $x$. However there is a shorter and more elegant way to approach this problem. Let's use structural features to rewrite the equation in a simpler form. Note that $4x^2+12=4({x^2+3})$. This means that we can rewrite the equation as: $({x^2+3})^2=4({x^2+3})$ If we let ${p}={x^2+3}$, we can see that this equation is in the form: ${p}^2=4{p}$ Let's solve this equation in terms of ${p}$ : $\begin{aligned}{p}^2&=4{p}\\\\ {p}^2-4{p}&=0\\\\ {p}({p}-4)&=0\\\\ {p}=0\ &\text{or} \ \ {p}=4 \end{aligned}$ Since ${p}={x^2+3}$, let's substitute this value back into our two solutions in order to solve for $x$ : ${x^2+3}=0\ \ \ \text{or} \ \ \ {x^2+3}=4$ Note that there are no real solutions to the equation ${x^2+3}=0$. [Why not?] When we solve ${x^2+3}=4$, we find that $x=\pm{1}$. In conclusion, the two solutions of the equation $(x^2+3)^2=4x^2+12$ are $x=1$ and $x=-1$. [Is there another way to solve for x?]